Some $800 million in federal money is on the line, as well as millions in state aid to local schools. On Tuesday, Gov. Andrew M. Cuomo put everyone on notice when he unveiled the details of his budget plan, ordering school districts to settle on a new teacher evaluation system by Jan. 17, 2013, or lose their share of a proposed 4 percent increase in education spending.

He gave the Regents and the teachers’ union 30 days to resolve their lawsuit. It is either that, he said, or adopt an evaluation system that he would impose.

The sides are not as far apart as their public posture would indicate. Three weeks before Mr. Cuomo set the deadline, the union had already acceded to one of the state’s key demands. It agreed that most of the 60 points teachers could earn on subjective measurements should be based on classroom observations — something the state’s education commissioner, John B. King Jr., had been pushing for. Of the total score of 100, results from student testing would account for the other 40 points.

The union’s president, Richard C. Iannuzzi, sounded optimistic on Wednesday, saying in an interview that a settlement could be reached “in two or three days.” Dr. King, however, said there were many differences to be resolved.

“Conversations are ongoing,” he said, “but there’s a distance to travel.”

Last week, the federal Education Department warned New York that it could lose the $700 million in education financing it was awarded last year as part of the Race to the Top program if it did not adopt a system to evaluate teachers and principals statewide, one of the program’s requirements. By then, Dr. King had already suspended a smaller pot of federal money — $58 million in grants to help struggling schools in New York City, as well as nine other school districts in the state. The districts and their unions should have reached agreements on an evaluation process by Dec. 31, as the grants stipulated, but did not.

Mr. Cuomo’s hard-line message aims to resolve both problems, in part by applying renewed pressure on the Bloomberg administration and the president of the city teachers’ union, Michael Mulgrew, to go back to the negotiating table. Talks collapsed days before the deadline established by the grants, and the sides were so far apart, city officials said, that there was no point in further discussion.

At issue is the process by which teachers would be able to appeal a poor rating. The city proposed forming a three-person committee consisting of one representative from the city, one from the union and one who would be jointly selected by both to issue an advisory decision to the schools chancellor, who would then make the final call.

Mr. Mulgrew objected. He said the administration had forced teachers to go to court to have bad ratings reversed.

“They’re now using our objections to say we’re obstructionists,” he said. The appeals process is the only obstacle to a compromise, he said.

The appeals process will be a crucial issue for the union once the teacher evaluation standards go into effect. The new law scraps a “satisfactory/unsatisfactory” scale that has been used to judge teachers for decades and introduces a four-tiered rating: “ineffective,” “developing,” “effective” and “highly effective.” Teachers who are rated “ineffective” for two consecutive years could lose their jobs within 60 days. Under the current system, less than 3 percent of the city’s teachers are rated “unsatisfactory,” and it can take more than a year to fire a teacher.

The Legislature approved the new system unanimously in June 2010. Union and education officials stood side by side in Albany to celebrate their joint achievement, and Mr. Mulgrew traveled to Washington to testify on behalf of the state’s application for the Race to the Top money.

The disagreements that remain are, by and large, the subjects of the lawsuit by the state teachers’ union. For example, 40 points on the annual reviews for teachers statewide would come from students’ test scores. The union wants only half of those points to be based on standardized tests, but the Board of Regents, which sets state education policy, allowed districts to base all of the 40 points on standardized tests.

The law specifies that 20 points of the evaluation must be based on the state tests and the remaining 20 points on other exams, to be developed by local districts. The discrepancy between the Regents’ regulations and the legislation is the reason the union sued, Mr. Iannuzzi said.

“We never challenged the law. We only challenged their interpretation of the law,” he said.

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